Solve The System Of Equations:${ \begin{align*} X + 2Y &= 5 \ 3X + 2Y &= 17 \end{align*} }$

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Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables, X and Y. We will use the given system of equations as an example and provide a step-by-step guide on how to solve it.

The System of Equations

The given system of equations is:

{ \begin{align*} X + 2Y &= 5 \\ 3X + 2Y &= 17 \end{align*} }

Method 1: Substitution Method

One way to solve this system of equations is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Step 1: Solve the First Equation for X

We can start by solving the first equation for X:

X+2Y=5{ X + 2Y = 5 }

Subtracting 2Y from both sides gives us:

X=5βˆ’2Y{ X = 5 - 2Y }

Step 2: Substitute the Expression for X into the Second Equation

Now, we can substitute the expression for X into the second equation:

3(5βˆ’2Y)+2Y=17{ 3(5 - 2Y) + 2Y = 17 }

Expanding the equation gives us:

15βˆ’6Y+2Y=17{ 15 - 6Y + 2Y = 17 }

Combining like terms gives us:

βˆ’4Y=2{ -4Y = 2 }

Dividing both sides by -4 gives us:

Y=βˆ’12{ Y = -\frac{1}{2} }

Step 3: Find the Value of X

Now that we have the value of Y, we can substitute it back into one of the original equations to find the value of X. We will use the first equation:

X+2Y=5{ X + 2Y = 5 }

Substituting Y = -1/2 gives us:

X+2(βˆ’12)=5{ X + 2(-\frac{1}{2}) = 5 }

Simplifying the equation gives us:

Xβˆ’1=5{ X - 1 = 5 }

Adding 1 to both sides gives us:

X=6{ X = 6 }

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to make the coefficients of either X or Y the same in both equations. We can multiply the first equation by 3 and the second equation by 1:

3(X+2Y)=3(5){ 3(X + 2Y) = 3(5) }

3X+6Y=15{ 3X + 6Y = 15 }

3X+2Y=17{ 3X + 2Y = 17 }

Step 2: Subtract the Second Equation from the First Equation

Now, we can subtract the second equation from the first equation to eliminate X:

(3X+6Y)βˆ’(3X+2Y)=15βˆ’17{ (3X + 6Y) - (3X + 2Y) = 15 - 17 }

Simplifying the equation gives us:

4Y=βˆ’2{ 4Y = -2 }

Dividing both sides by 4 gives us:

Y=βˆ’12{ Y = -\frac{1}{2} }

Step 3: Find the Value of X

Now that we have the value of Y, we can substitute it back into one of the original equations to find the value of X. We will use the first equation:

X+2Y=5{ X + 2Y = 5 }

Substituting Y = -1/2 gives us:

X+2(βˆ’12)=5{ X + 2(-\frac{1}{2}) = 5 }

Simplifying the equation gives us:

Xβˆ’1=5{ X - 1 = 5 }

Adding 1 to both sides gives us:

X=6{ X = 6 }

Conclusion

In this article, we have solved a system of two linear equations with two variables, X and Y, using the substitution method and the elimination method. We have shown that both methods can be used to solve the system of equations and have provided a step-by-step guide on how to do it. The solution to the system of equations is X = 6 and Y = -1/2.

Applications

Solving systems of linear equations has many applications in mathematics, science, and engineering. Some of the applications include:

  • Linear Algebra: Solving systems of linear equations is a fundamental concept in linear algebra, which is used to study the properties of linear transformations and vector spaces.
  • Computer Science: Solving systems of linear equations is used in computer science to solve problems in computer graphics, machine learning, and data analysis.
  • Physics: Solving systems of linear equations is used in physics to solve problems in mechanics, electromagnetism, and thermodynamics.
  • Engineering: Solving systems of linear equations is used in engineering to solve problems in structural analysis, circuit analysis, and control systems.

Final Thoughts

Solving systems of linear equations is a fundamental concept in mathematics that has many applications in science and engineering. In this article, we have provided a step-by-step guide on how to solve a system of two linear equations with two variables using the substitution method and the elimination method. We have shown that both methods can be used to solve the system of equations and have provided a solution to the system of equations. We hope that this article has provided a useful resource for students and professionals who need to solve systems of linear equations.

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Q: What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

Q: What are the two main methods for solving systems of linear equations?

The two main methods for solving systems of linear equations are the substitution method and the elimination method.

Q: What is the substitution method?

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.

Q: What is the elimination method?

The elimination method involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I know which method to use?

You can use either method, but the substitution method is often easier to use when one of the equations is already solved for one variable. The elimination method is often easier to use when the coefficients of one of the variables are the same in both equations.

Q: What if I have a system of three or more linear equations?

If you have a system of three or more linear equations, you can use the elimination method to eliminate one of the variables and then use the substitution method to solve for the remaining variables.

Q: Can I use a calculator to solve systems of linear equations?

Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions for solving systems of linear equations.

Q: What if I get a solution that doesn't make sense?

If you get a solution that doesn't make sense, it may be because you made a mistake in solving the system of equations. Check your work carefully and make sure that you are using the correct method.

Q: Can I use systems of linear equations to solve real-world problems?

Yes, you can use systems of linear equations to solve real-world problems. Many problems in science, engineering, and economics can be modeled using systems of linear equations.

Q: What are some examples of real-world problems that can be solved using systems of linear equations?

Some examples of real-world problems that can be solved using systems of linear equations include:

  • Finance: Solving systems of linear equations can be used to calculate interest rates, investment returns, and other financial metrics.
  • Science: Solving systems of linear equations can be used to model population growth, chemical reactions, and other scientific phenomena.
  • Engineering: Solving systems of linear equations can be used to design bridges, buildings, and other structures.
  • Economics: Solving systems of linear equations can be used to model economic systems, including supply and demand, inflation, and unemployment.

Q: What are some common mistakes to avoid when solving systems of linear equations?

Some common mistakes to avoid when solving systems of linear equations include:

  • Not checking your work: Make sure to check your work carefully to avoid mistakes.
  • Not using the correct method: Make sure to use the correct method for the problem you are trying to solve.
  • Not simplifying the equations: Make sure to simplify the equations as much as possible to avoid confusion.
  • Not checking for extraneous solutions: Make sure to check for extraneous solutions, which are solutions that do not make sense in the context of the problem.

Q: How can I practice solving systems of linear equations?

You can practice solving systems of linear equations by working through examples and exercises in a textbook or online resource. You can also try solving real-world problems that involve systems of linear equations.

Q: What are some resources for learning more about solving systems of linear equations?

Some resources for learning more about solving systems of linear equations include:

  • Textbooks: There are many textbooks available that cover solving systems of linear equations, including "Linear Algebra and Its Applications" by Gilbert Strang and "Introduction to Linear Algebra" by Jim Hefferon.
  • Online resources: There are many online resources available that cover solving systems of linear equations, including Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
  • Video tutorials: There are many video tutorials available that cover solving systems of linear equations, including 3Blue1Brown, Crash Course, and Math Antics.